Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1

Percentage Accurate: 66.3% → 99.7%

Time: 10.3s

Alternatives: 9

Speedup: 0.8×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

C

double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))

C

double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * x) / (y * y)) + ((z * z) / (t * t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * x) / (y * y)) + ((z * z) / (t * t));
}

Python

def code(x, y, z, t):
	return ((x * x) / (y * y)) + ((z * z) / (t * t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * x) / Float64(y * y)) + Float64(Float64(z * z) / Float64(t * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * x) / (y * y)) + ((z * z) / (t * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \frac{x}{y}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (/ z t) (/ z t) (* (/ x y) (/ x y))))

C

double code(double x, double y, double z, double t) {
	return fma((z / t), (z / t), ((x / y) * (x / y)));
}

Julia

function code(x, y, z, t)
	return fma(Float64(z / t), Float64(z / t), Float64(Float64(x / y) * Float64(x / y)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

   3. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{z \cdot z}}{t \cdot t} + \frac{x \cdot x}{y \cdot y} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} + \frac{x \cdot x}{y \cdot y} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)} \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t}}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right) \]

   9. lower-/.f64 78.8

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{\frac{z}{t}}, \frac{x \cdot x}{y \cdot y}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{x \cdot x}}{y \cdot y}\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{\color{blue}{y \cdot y}}\right) \]

   13. times-frac N/A

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

   14. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

   15. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

   16. lower-/.f64 99.7

       \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\color{blue}{\left(\frac{x}{y}\right)}}^{2}\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

   2. pow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

   3. lift-*.f64 99.7

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]
7. Add Preprocessing

Alternative 2: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ t_2 := \frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{if}\;t\_1 \leq 10^{-137}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+215}:\\ \;\;\;\;t\_1 + \frac{z \cdot z}{t \cdot t}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{\left(x \cdot x\right) \cdot t}{y}}{y}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))) (t_2 (* (/ z t) (/ z t))))
   (if (<= t_1 1e-137)
     t_2
     (if (<= t_1 2e+215)
       (+ t_1 (/ (* z z) (* t t)))
       (if (<= t_1 INFINITY) (/ (/ (/ (* (* x x) t) y) y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) * (z / t);
	double tmp;
	if (t_1 <= 1e-137) {
		tmp = t_2;
	} else if (t_1 <= 2e+215) {
		tmp = t_1 + ((z * z) / (t * t));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((((x * x) * t) / y) / y) / t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double t_2 = (z / t) * (z / t);
	double tmp;
	if (t_1 <= 1e-137) {
		tmp = t_2;
	} else if (t_1 <= 2e+215) {
		tmp = t_1 + ((z * z) / (t * t));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((((x * x) * t) / y) / y) / t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	t_2 = (z / t) * (z / t)
	tmp = 0
	if t_1 <= 1e-137:
		tmp = t_2
	elif t_1 <= 2e+215:
		tmp = t_1 + ((z * z) / (t * t))
	elif t_1 <= math.inf:
		tmp = ((((x * x) * t) / y) / y) / t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	t_2 = Float64(Float64(z / t) * Float64(z / t))
	tmp = 0.0
	if (t_1 <= 1e-137)
		tmp = t_2;
	elseif (t_1 <= 2e+215)
		tmp = Float64(t_1 + Float64(Float64(z * z) / Float64(t * t)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * t) / y) / y) / t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	t_2 = (z / t) * (z / t);
	tmp = 0.0;
	if (t_1 <= 1e-137)
		tmp = t_2;
	elseif (t_1 <= 2e+215)
		tmp = t_1 + ((z * z) / (t * t));
	elseif (t_1 <= Inf)
		tmp = ((((x * x) * t) / y) / y) / t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-137], t$95$2, If[LessEqual[t$95$1, 2e+215], N[(t$95$1 + N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision] / t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 9.99999999999999978e-138 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 50.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      4. unpow2 N/A

         \[\leadsto \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      7. lower-/.f64 75.6

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

   5. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 79.8%

      \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   if 9.99999999999999978e-138 < (/.f64 (*.f64 x x) (*.f64 y y)) < 1.99999999999999981e215

   1. Initial program 89.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   if 1.99999999999999981e215 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 71.2%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t}}{t}} + \frac{x \cdot x}{y \cdot y} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \color{blue}{\frac{x \cdot x}{y \cdot y}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \frac{x \cdot x}{\color{blue}{y \cdot y}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \color{blue}{\frac{\frac{x \cdot x}{y}}{y}} \]

      9. frac-add N/A

         \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t} \cdot y + t \cdot \frac{x \cdot x}{y}}{t \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t} \cdot y + t \cdot \frac{x \cdot x}{y}}{t \cdot y}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z \cdot z}{t}, y, t \cdot \frac{x \cdot x}{y}\right)}}{t \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{z \cdot z}}{t}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      13. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t}} \cdot z, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, \color{blue}{t \cdot \frac{x \cdot x}{y}}\right)}{t \cdot y} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \frac{\color{blue}{x \cdot x}}{y}\right)}{t \cdot y} \]

      18. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\color{blue}{\frac{x}{y}} \cdot x\right)\right)}{t \cdot y} \]

      21. lower-*.f64 88.6

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{\color{blue}{t \cdot y}} \]

   4. Applied rewrites 88.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{t \cdot y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y \cdot {z}^{2}}{t}}}{t \cdot y} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot {z}^{2}}{t}}}{t \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{z}^{2} \cdot y}}{t}}{t \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{{z}^{2} \cdot y}}{t}}{t \cdot y} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{t}}{t \cdot y} \]

      5. lower-*.f64 32.1

         \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{t}}{t \cdot y} \]

   7. Applied rewrites 32.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(z \cdot z\right) \cdot y}{t}}}{t \cdot y} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(z \cdot z\right) \cdot y}{t}}{t \cdot y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\left(z \cdot z\right) \cdot y}{t}}{\color{blue}{t \cdot y}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\left(z \cdot z\right) \cdot y}{t}}{\color{blue}{y \cdot t}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(z \cdot z\right) \cdot y}{t}}{y}}{t}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\frac{\left(z \cdot z\right) \cdot y}{t}}{y}}{t}} \]

   9. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{\frac{\left(y \cdot \frac{z}{t}\right) \cdot z}{y}}{t}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\color{blue}{\frac{t \cdot {x}^{2}}{{y}^{2}}}}{t} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\frac{t \cdot {x}^{2}}{\color{blue}{y \cdot y}}}{t} \]

       2. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{t \cdot {x}^{2}}{y}}{y}}}{t} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{t \cdot {x}^{2}}{y}}{y}}}{t} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{t \cdot {x}^{2}}{y}}}{y}}{t} \]

       5. *-commutative N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{{x}^{2} \cdot t}}{y}}{y}}{t} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{{x}^{2} \cdot t}}{y}}{y}}{t} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot t}{y}}{y}}{t} \]

       8. lower-*.f64 83.1

          \[\leadsto \frac{\frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot t}{y}}{y}}{t} \]

   12. Applied rewrites 83.1%

       \[\leadsto \frac{\color{blue}{\frac{\frac{\left(x \cdot x\right) \cdot t}{y}}{y}}}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{+207}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e+207)
     (+ (* (/ x y) (/ x y)) t_1)
     (if (<= t_1 INFINITY)
       (* (/ z t) (/ z t))
       (fma (/ z t) (/ z t) (/ (* x x) (* y y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+207) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = fma((z / t), (z / t), ((x * x) / (y * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e+207)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(Float64(x * x) / Float64(y * y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+207], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1e207

   1. Initial program 71.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t} \]

      7. lower-/.f64 96.8

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   4. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   if 1e207 < (/.f64 (*.f64 z z) (*.f64 t t)) < +inf.0

   1. Initial program 74.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      4. unpow2 N/A

         \[\leadsto \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      7. lower-/.f64 91.9

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

   5. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.3%

      \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   if +inf.0 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 0.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{t \cdot t} + \frac{x \cdot x}{y \cdot y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} + \frac{x \cdot x}{y \cdot y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t}}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right) \]

      9. lower-/.f64 89.0

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{\frac{z}{t}}, \frac{x \cdot x}{y \cdot y}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{x \cdot x}}{y \cdot y}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{\color{blue}{y \cdot y}}\right) \]

      13. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      15. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      16. lower-/.f64 99.5

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\color{blue}{\left(\frac{x}{y}\right)}}^{2}\right) \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y}} \cdot \frac{x}{y}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \color{blue}{\frac{x}{y}}\right) \]

      5. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{x \cdot x}}{y \cdot y}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{\color{blue}{y \cdot y}}\right) \]

      8. lift-/.f64 89.0

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

   6. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+62} \lor \neg \left(t\_1 \leq \infty\right):\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot x\right) \cdot t}{y}}{t \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (or (<= t_1 5e+62) (not (<= t_1 INFINITY)))
     (* (/ z t) (/ z t))
     (/ (/ (* (* x x) t) y) (* t y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if ((t_1 <= 5e+62) || !(t_1 <= ((double) INFINITY))) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (((x * x) * t) / y) / (t * y);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if ((t_1 <= 5e+62) || !(t_1 <= Double.POSITIVE_INFINITY)) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (((x * x) * t) / y) / (t * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * x) / (y * y)
	tmp = 0
	if (t_1 <= 5e+62) or not (t_1 <= math.inf):
		tmp = (z / t) * (z / t)
	else:
		tmp = (((x * x) * t) / y) / (t * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if ((t_1 <= 5e+62) || !(t_1 <= Inf))
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * t) / y) / Float64(t * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * x) / (y * y);
	tmp = 0.0;
	if ((t_1 <= 5e+62) || ~((t_1 <= Inf)))
		tmp = (z / t) * (z / t);
	else
		tmp = (((x * x) * t) / y) / (t * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e+62], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision] / N[(t * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < 5.00000000000000029e62 or +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 54.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      4. unpow2 N/A

         \[\leadsto \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      7. lower-/.f64 73.6

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

   5. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.8%

      \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   if 5.00000000000000029e62 < (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 74.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t}}{t}} + \frac{x \cdot x}{y \cdot y} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \color{blue}{\frac{x \cdot x}{y \cdot y}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \frac{x \cdot x}{\color{blue}{y \cdot y}} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{z \cdot z}{t}}{t} + \color{blue}{\frac{\frac{x \cdot x}{y}}{y}} \]

      9. frac-add N/A

         \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t} \cdot y + t \cdot \frac{x \cdot x}{y}}{t \cdot y}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{z \cdot z}{t} \cdot y + t \cdot \frac{x \cdot x}{y}}{t \cdot y}} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{z \cdot z}{t}, y, t \cdot \frac{x \cdot x}{y}\right)}}{t \cdot y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{z \cdot z}}{t}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      13. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t} \cdot z}, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      15. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{z}{t}} \cdot z, y, t \cdot \frac{x \cdot x}{y}\right)}{t \cdot y} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, \color{blue}{t \cdot \frac{x \cdot x}{y}}\right)}{t \cdot y} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \frac{\color{blue}{x \cdot x}}{y}\right)}{t \cdot y} \]

      18. associate-*l/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)}\right)}{t \cdot y} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\color{blue}{\frac{x}{y}} \cdot x\right)\right)}{t \cdot y} \]

      21. lower-*.f64 87.8

          \[\leadsto \frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{\color{blue}{t \cdot y}} \]

   4. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{z}{t} \cdot z, y, t \cdot \left(\frac{x}{y} \cdot x\right)\right)}{t \cdot y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y \cdot {z}^{2}}{t}}}{t \cdot y} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot {z}^{2}}{t}}}{t \cdot y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{z}^{2} \cdot y}}{t}}{t \cdot y} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{{z}^{2} \cdot y}}{t}}{t \cdot y} \]

      4. unpow2 N/A

         \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{t}}{t \cdot y} \]

      5. lower-*.f64 31.1

         \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot z\right)} \cdot y}{t}}{t \cdot y} \]

   7. Applied rewrites 31.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(z \cdot z\right) \cdot y}{t}}}{t \cdot y} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.3%

      \[\leadsto \frac{z \cdot \color{blue}{\left(z \cdot \frac{y}{t}\right)}}{t \cdot y} \]

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\color{blue}{\frac{t \cdot {x}^{2}}{y}}}{t \cdot y} \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{t \cdot {x}^{2}}{y}}}{t \cdot y} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot t}}{y}}{t \cdot y} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot t}}{y}}{t \cdot y} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot t}{y}}{t \cdot y} \]

       5. lower-*.f64 81.8

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot t}{y}}{t \cdot y} \]

   12. Applied rewrites 81.8%

       \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot x\right) \cdot t}{y}}}{t \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x}{y \cdot y} \leq 5 \cdot 10^{+62} \lor \neg \left(\frac{x \cdot x}{y \cdot y} \leq \infty\right):\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot x\right) \cdot t}{y}}{t \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, x \cdot \frac{x}{y \cdot y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 5e+187)
     (+ (* (/ x y) (/ x y)) t_1)
     (fma (/ z t) (/ z t) (* x (/ x (* y y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 5e+187) {
		tmp = ((x / y) * (x / y)) + t_1;
	} else {
		tmp = fma((z / t), (z / t), (x * (x / (y * y))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 5e+187)
		tmp = Float64(Float64(Float64(x / y) * Float64(x / y)) + t_1);
	else
		tmp = fma(Float64(z / t), Float64(z / t), Float64(x * Float64(x / Float64(y * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+187], N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 5.0000000000000001e187

   1. Initial program 71.4%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t} \]

      7. lower-/.f64 96.8

         \[\leadsto \frac{x}{y} \cdot \color{blue}{\frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   4. Applied rewrites 96.8%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + \frac{z \cdot z}{t \cdot t} \]

   if 5.0000000000000001e187 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 51.8%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{t \cdot t} + \frac{x \cdot x}{y \cdot y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} + \frac{x \cdot x}{y \cdot y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t}}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right) \]

      9. lower-/.f64 84.2

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{\frac{z}{t}}, \frac{x \cdot x}{y \cdot y}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{x \cdot x}}{y \cdot y}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{\color{blue}{y \cdot y}}\right) \]

      13. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      15. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      16. lower-/.f64 99.8

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\color{blue}{\left(\frac{x}{y}\right)}}^{2}\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y}} \cdot \frac{x}{y}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \color{blue}{\frac{x}{y}}\right) \]

      5. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

      6. sqr-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}{y \cdot y}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}{\color{blue}{y \cdot y}}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\mathsf{neg}\left(x\right)}{y \cdot y}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\mathsf{neg}\left(x\right)}{y \cdot y}}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\left(-x\right)} \cdot \frac{\mathsf{neg}\left(x\right)}{y \cdot y}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{\mathsf{neg}\left(x\right)}{\color{blue}{y \cdot y}}\right) \]

      12. sqr-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}}\right) \]

      13. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\mathsf{neg}\left(y \cdot \left(\mathsf{neg}\left(y\right)\right)\right)}}\right) \]

      14. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y \cdot y\right)\right)}\right)}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{y \cdot y}\right)\right)\right)}\right) \]

      16. frac-2neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \color{blue}{\frac{x}{\mathsf{neg}\left(y \cdot y\right)}}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \color{blue}{\frac{x}{\mathsf{neg}\left(y \cdot y\right)}}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{x}{\mathsf{neg}\left(\color{blue}{y \cdot y}\right)}\right) \]

      19. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{x}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{x}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y}}\right) \]

      21. lower-neg.f64 94.3

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \left(-x\right) \cdot \frac{x}{\color{blue}{\left(-y\right)} \cdot y}\right) \]

   6. Applied rewrites 94.3%

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\left(-x\right) \cdot \frac{x}{\left(-y\right) \cdot y}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{y} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, x \cdot \frac{x}{y \cdot y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot x}{y \cdot y}\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x x) (* y y))))
   (if (<= t_1 INFINITY) (fma (/ z t) (/ z t) t_1) (* (/ z t) (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * x) / (y * y);
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = fma((z / t), (z / t), t_1);
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * x) / Float64(y * y))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = fma(Float64(z / t), Float64(z / t), t_1);
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 72.1%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t} + \frac{x \cdot x}{y \cdot y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot z}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{t \cdot t} + \frac{x \cdot x}{y \cdot y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot z}{\color{blue}{t \cdot t}} + \frac{x \cdot x}{y \cdot y} \]

      6. times-frac N/A

         \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{z}{t}} + \frac{x \cdot x}{y \cdot y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{t}}, \frac{z}{t}, \frac{x \cdot x}{y \cdot y}\right) \]

      9. lower-/.f64 91.3

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \color{blue}{\frac{z}{t}}, \frac{x \cdot x}{y \cdot y}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{x \cdot x}}{y \cdot y}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{\color{blue}{y \cdot y}}\right) \]

      13. times-frac N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

      14. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      15. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      16. lower-/.f64 99.7

          \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\color{blue}{\left(\frac{x}{y}\right)}}^{2}\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, {\left(\frac{x}{y}\right)}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x}{y}} \cdot \frac{x}{y}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x}{y} \cdot \color{blue}{\frac{x}{y}}\right) \]

      5. frac-times N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{\color{blue}{x \cdot x}}{y \cdot y}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \frac{x \cdot x}{\color{blue}{y \cdot y}}\right) \]

      8. lift-/.f64 91.3

         \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

   6. Applied rewrites 91.3%

      \[\leadsto \mathsf{fma}\left(\frac{z}{t}, \frac{z}{t}, \color{blue}{\frac{x \cdot x}{y \cdot y}}\right) \]

   if +inf.0 < (/.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 0.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      4. unpow2 N/A

         \[\leadsto \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      7. lower-/.f64 51.6

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 51.6%

      \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot z}{t \cdot t}\\ \mathbf{if}\;t\_1 \leq 10^{+207}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z z) (* t t))))
   (if (<= t_1 1e+207) (+ (* x (/ x (* y y))) t_1) (* (/ z t) (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+207) {
		tmp = (x * (x / (y * y))) + t_1;
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * z) / (t * t)
    if (t_1 <= 1d+207) then
        tmp = (x * (x / (y * y))) + t_1
    else
        tmp = (z / t) * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * z) / (t * t);
	double tmp;
	if (t_1 <= 1e+207) {
		tmp = (x * (x / (y * y))) + t_1;
	} else {
		tmp = (z / t) * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * z) / (t * t)
	tmp = 0
	if t_1 <= 1e+207:
		tmp = (x * (x / (y * y))) + t_1
	else:
		tmp = (z / t) * (z / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * z) / Float64(t * t))
	tmp = 0.0
	if (t_1 <= 1e+207)
		tmp = Float64(Float64(x * Float64(x / Float64(y * y))) + t_1);
	else
		tmp = Float64(Float64(z / t) * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * z) / (t * t);
	tmp = 0.0;
	if (t_1 <= 1e+207)
		tmp = (x * (x / (y * y))) + t_1;
	else
		tmp = (z / t) * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+207], N[(N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z z) (*.f64 t t)) < 1e207

   1. Initial program 71.6%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot x}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      3. sqr-neg-rev N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\mathsf{neg}\left(x\right)}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\mathsf{neg}\left(x\right)}{y \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{\mathsf{neg}\left(x\right)}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]

      7. frac-2neg N/A

         \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)}{\mathsf{neg}\left(y \cdot y\right)}} + \frac{z \cdot z}{t \cdot t} \]

      8. remove-double-neg N/A

         \[\leadsto \left(-x\right) \cdot \frac{\color{blue}{x}}{\mathsf{neg}\left(y \cdot y\right)} + \frac{z \cdot z}{t \cdot t} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{x}{\mathsf{neg}\left(y \cdot y\right)}} + \frac{z \cdot z}{t \cdot t} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x\right) \cdot \frac{x}{\mathsf{neg}\left(\color{blue}{y \cdot y}\right)} + \frac{z \cdot z}{t \cdot t} \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(-x\right) \cdot \frac{x}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(-x\right) \cdot \frac{x}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

      13. lower-neg.f64 79.0

          \[\leadsto \left(-x\right) \cdot \frac{x}{\color{blue}{\left(-y\right)} \cdot y} + \frac{z \cdot z}{t \cdot t} \]

   4. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{x}{\left(-y\right) \cdot y}} + \frac{z \cdot z}{t \cdot t} \]

   if 1e207 < (/.f64 (*.f64 z z) (*.f64 t t))

   1. Initial program 51.3%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      4. unpow2 N/A

         \[\leadsto \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      7. lower-/.f64 78.8

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.4%

      \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot z}{t \cdot t} \leq 10^{+207}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot y} + \frac{z \cdot z}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{z}{t} \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t \cdot t} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 2.5e+110) (* (/ z t) (/ z t)) (* (/ z (* t t)) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.5e+110) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= 2.5d+110) then
        tmp = (z / t) * (z / t)
    else
        tmp = (z / (t * t)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 2.5e+110) {
		tmp = (z / t) * (z / t);
	} else {
		tmp = (z / (t * t)) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= 2.5e+110:
		tmp = (z / t) * (z / t)
	else:
		tmp = (z / (t * t)) * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 2.5e+110)
		tmp = Float64(Float64(z / t) * Float64(z / t));
	else
		tmp = Float64(Float64(z / Float64(t * t)) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 2.5e+110)
		tmp = (z / t) * (z / t);
	else
		tmp = (z / (t * t)) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 2.5e+110], N[(N[(z / t), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.49999999999999989e110

   1. Initial program 62.5%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      4. unpow2 N/A

         \[\leadsto \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      7. lower-/.f64 56.2

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.2%

      \[\leadsto \frac{z}{t} \cdot \color{blue}{\frac{z}{t}} \]

   if 2.49999999999999989e110 < x

   1. Initial program 61.0%

      \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

      2. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

      4. unpow2 N/A

         \[\leadsto \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

      7. lower-/.f64 51.8

         \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.1%

      \[\leadsto \frac{z}{t \cdot t} \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{z}{t \cdot t} \cdot z \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ z (* t t)) z))

C

double code(double x, double y, double z, double t) {
	return (z / (t * t)) * z;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (z / (t * t)) * z
end function

Java

public static double code(double x, double y, double z, double t) {
	return (z / (t * t)) * z;
}

Python

def code(x, y, z, t):
	return (z / (t * t)) * z

Julia

function code(x, y, z, t)
	return Float64(Float64(z / Float64(t * t)) * z)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (z / (t * t)) * z;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / N[(t * t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 62.3%

   \[\frac{x \cdot x}{y \cdot y} + \frac{z \cdot z}{t \cdot t} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{{z}^{2}}{{t}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{z \cdot z}}{{t}^{2}} \]

   2. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{z}{{t}^{2}} \cdot z} \]

   4. unpow2 N/A

      \[\leadsto \frac{z}{\color{blue}{t \cdot t}} \cdot z \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t}} \cdot z \]

   7. lower-/.f64 55.5

      \[\leadsto \frac{\color{blue}{\frac{z}{t}}}{t} \cdot z \]

5. Applied rewrites 55.5%

   \[\leadsto \color{blue}{\frac{\frac{z}{t}}{t} \cdot z} \]
6. Step-by-step derivation

7. Applied rewrites 50.6%

   \[\leadsto \frac{z}{t \cdot t} \cdot z \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x}{y}\right)}^{2} + {\left(\frac{z}{t}\right)}^{2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (pow (/ x y) 2.0) (pow (/ z t) 2.0)))

C

double code(double x, double y, double z, double t) {
	return pow((x / y), 2.0) + pow((z / t), 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / y) ** 2.0d0) + ((z / t) ** 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow((x / y), 2.0) + Math.pow((z / t), 2.0);
}

Python

def code(x, y, z, t):
	return math.pow((x / y), 2.0) + math.pow((z / t), 2.0)

Julia

function code(x, y, z, t)
	return Float64((Float64(x / y) ^ 2.0) + (Float64(z / t) ^ 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x / y) ^ 2.0) + ((z / t) ^ 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(z / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024329 
(FPCore (x y z t)
  :name "Graphics.Rasterific.Svg.PathConverter:arcToSegments from rasterific-svg-0.2.3.1"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (pow (/ x y) 2) (pow (/ z t) 2)))

  (+ (/ (* x x) (* y y)) (/ (* z z) (* t t))))